Download The kinetic theory of electromagnetic radiation

The following paper appeared in Speculations in Science and Technology, Vol.
8, No. 4, pages 263–272.
The kinetic theory of electromagnetic radiation
C. K. Thornhill
39 Crofton Road, Orpington, Kent BR6 8AE, UK
Received: August 1983
Abstract It is shown that Planck’s energy distribution for a black-body radiation field can
be simply derived for a gas-like ether with Maxwellian statistics. The gas consists of an infinite
variety of particles, whose masses are integral multiples n of the mass of the unit particle,
the abundance of n-particles being proportional to n−4 . The frequency of electromagnetic
waves correlates with the energy per unit mass of the particles, not with their energy, thus
differing from Planck’s quantum hypothesis. Identifying the special wave-speed, usually called
the speed of light, with the wave-speed in the 2.7o K background radiation field, leads to a
mass 12 × 10−39 (kg) for the unit ether-particle, and an average number of about 360 ether
particles per cubic centimetre in the background radiation field, whose density is about 0.2 ×
10−30 (kg)/m3 .
’There fields of light and liquid ether flow’ (Dryden).
1 Introduction
The question, whether or not there is a physical ethereal medium in which
electromagnetic waves propagate, has been asked for many centuries. On the
one hand, there have always been those who have maintained that it is not
a sensible question to ask, since radiation is observed to have many physical
properties and cannot, therefore, exist in a true vacuum or void which is, by
definition, the total absence of anything physical. On the other hand, for about
the last hundred years, it has come to be largely accepted that there is no
physical ethereal medium, and the physical properties of radiation have been
transmogrified into waves, and energy parcels or photons, in a space-time metric.
The arguments for the denial of a physical ethereal medium are manifold
(see, for example, Whittaker1 ). One of these asserts that Maxwell’s equations
show that electromagnetic waves are transverse and that, therefore, any ethereal medium must behave like an elastic solid. This argument is invalid, since
Maxwell’s equations only show that the oscillating electric and magnetic fields
are transverse to the direction of wave propagation, and can say nothing whatsoever about any condensational oscillations of any possible physical medium
in which the waves are propagating. In fact, the deduction, from Maxwell’s
equations, that electromagnetic waves are entirely transverse, is no more than
a restatement of an assumption that there is no physical ethereal medium. On
the contrary, if there is such a medium, one would deduce from Maxwell’s equa1
tions, since electric field, magnetic field and motion are mutually perpendicular
for plane waves, that its condensational oscillations are longitudinal, in exact
analogy with sound waves in a fluid.
Another argument against the existence of a physical ethereal medium is
that Planck’s empirical formula, for the energy distribution in a black-body radiation field, cannot be derived from the kinetic theory of a gas with Maxwellian
statistics. Indeed, it is well-known that kinetic theory and Maxwellian statistics
lead to an energy distribution which is a sum of Wien-type distributions, for a
gas mixture with any number of different kinds of atoms or molecules. But this
only establishes the impossibility of so deriving Planck’s distribution for a gas
with a finite variety of atoms or molecules. To assert the complete impossibility
of so deriving Planck’s distribution it is essential to eliminate the case of a gas
with an infinite variety of atoms or molecules, i .e . infinite in a mathematical
sense, but physically, in practice, a very large variety. The burden of the present
paper is to show that this possibility cannot be eliminated, but rather that it
permits a far simpler derivation of Planck’s energy distribution than has been
given anywhere heretofore.
2 Ethereal thermodynamics
If the ethereal medium is particulate like a gas, the black-body state may be
taken to correspond with thermodynamic equilibrium in a gas. Observation
(or, perhaps, lack of observation) then indicates that the size and mass of ether
particles must be at least orders of magnitude lower than the size and mass of
even the fundamental particles, and suggests, therefore, that such a medium
may be expected to behave like an ideal gas.
A well-known property of black-body radiation gives, for the energy density
E
= A0 T 4
v
(2.1)
where A0 is called the radiation density ’constant’. (Here, the notation used
is appropriate to fluid thermodynamics, namely pressure p, specific volume v,
specific entropy S, temperature T , intrinsic energy per unit mass E; so that
energy density, i .e . intrinsic energy per unit volume, becomes E/v.)
The thermodynamic properties of an ideal gas are completely specified by
the expression for E as a function of v and S, namely
Kcv
S
E = ω−1
exp
(2.2)
v
cv
where K, cv , and ω are constants. (The E, v, S system of thermodynamics is
used here, in which E is the primary dependent variable, v and S the independent variables. Partial differentiation is confined to v and S, so that suffices v
and S may be used to denote partial derivatives.) (See, for example, Thornhill2
or Swan and Thornhill3 .)
The thermodynamic identities γ ≡ vEvv /Ev , and cv ≡ ES /ESS show that
ω is the constant first adiabatic index, and cv is the constant specific heat at
2
constant volume. The temperature T is given by
K
S
T ≡ ES = ω−1 exp
v
cv
(2.3)
The relations (2.2) and (2.3) yield very simply,
−1
(ω−1)
ω
E
S
T (ω−1)
= cv K exp
v
cv
(2.4)
It is now clear that Eqs (2.1) and (2.4) are equivalent if the ’constancy’ of
A0 is attributed to a constancy of entropy in the black-body radiation fields to
which Eq.(2.1) is being applied. And, if this is done, it then follows that the
first adiabatic index ω of the ether, and the number of degrees of freedom α of
ether particles, are given by
ω
= 4,
ω−1
whence
ω=
4
3
and
(α + 2)
, whence α = 6
α
Thus, the quest for a gas-like ethereal medium, satisfying Planck’s form for
the energy distribution, is directed to an ideal gas formed by an infinite variety
of particles, all having six degrees of freedom.
ω=
3 Kinetic theory
The simplest and most obvious approach to the problem under consideration is
first to choose an infinite variety of ether particles, and then to try to determine
a mixture of them which will yield Planck’s energy distribution. A clue to the
choice of infinite variety comes directly from observation of the photoelectric effect, for this indicates that, in interactions between matter and radiation, energy
exchanges occur, at any frequency, ν, in integral multiples of some minimum
quantity, h0 ν. This suggests the choice of a single infinity of ether particles,
whose masses are integral multiples of some minimum mass m, but which, for
the purposes of kinetic theory, at least, may be considered as otherwise identical.
Consider then, a gas, occupying a volume V , whose particles all have three
degrees of freedom of translational motion, and three other degrees of freedom;
and whose particles are all identical except for (size? and) mass, their masses
being integral multiples n, (n = 1, ..., ∞) of an absolute mass quantum m, the
mass of a unit particle.
Let denote energy per unit mass, Nn () the number of n-particles in the
range (0, ), N n the total number of n-particles, and c2n the mean value of the
square of the translational speed of n-particles. Then, for Maxwellian statistics
(see Appendix),
−3
∂Nn () 27N n 2
exp
(3.1)
=
2c6n
c2n
∂
3
whilst, for equi-partition of energy, i .e . all six degrees of freedom of all particles,
whatever their masses, have the same mean energy, mnc2n must be the same for
all n, so that temperature T may be defined by
mnc2n
= kT
3
(3.2)
where k is a universal constant.
If, now E(, T ) denotes the total energy of all particles in the range (0, ),
then
∞
∞ X
X
−mn
∂Nn ()
N n m4 n4 3
∂E(, T )
exp
=
=
(3.3)
mn
∂
∂
2k 3 T 3
kT
n=1
n=1
A mixture of the particles must now be specified, i .e . an abundance function N n
must be specified, and then the summation in Eq. (3.3) can be performed and
the resulting expression for ∂E/∂ compared with Planck’s energy distribution.
But it is, by now, already obvious that the simplest choice for N n , namely
N n ∝ 1/n4 , will succeed.
For, if N n = δ/n4 , then the total number of all the particles, N , is given by
N=
∞
X
δ
π4 δ
=
4
n
90
n=1
(3.4)
and the total mass of the gas, denoted by N m, is given by
Nm =
∞
X
mδ
= mδζ(3)
n3
n=1
(3.5)
where ζ denotes Riemann’s zeta-function. Thus,
δ=
N
ζ(3)
(3.6)
and the mean mass m of all the particles satisfies
m=
N m 90ζ(3)m
=
N
π4
(3.7)
Substituting these values in Eq. (3.3) then leads to
"
#
∞
X
m4 N
−mn
∂E(, T )
3
=
exp
∂
2k 3 T 3 ζ(3)
kT
n=1
whence
∂E(, T )
=
∂
"
#
m4 N
3
2k 3 T 3 ζ(3) exp m
kT − 1
4
(3.8)
The total energy of the gas is
∞
Z
E(∞, T ) =
0
π 4 N kT
∂E(, T )
d =
∂
30ζ(3)
and the mean energy per unit mass is thus
π4
kT 3kT 9c2
E(∞, T )
=
= c2
=
=
E=
m
4
30ζ(3) m
Nm
(3.9)
(3.10)
where c is the wave speed, and c is the r.m.s. translational speed of all the
particles. So
3c
(3.11)
c=
2
4 Black-body radiation
For black-body radiation, E is found, by observation, to depend on the temperature T , and on the frequency ν of electromagnetic waves, in accordance with
the empirical relation first suggested by Planck, namely
1 ∂E(ν, T )
8πh0
ν3
h
i
=
(4.1)
3
h0 ν
V
∂ν
c0
exp kT
−1
where h0 is called Planck’s ’constant’ and c0 is called the ’speed of light’.
In order to reconcile Eq. (4.1) with the above result, Eq. (3.8), two conditions must be satisfied. they are
=
and
h0 ν
m
(4.2)
N 16πk 3 T 3 ζ(3)
=
V
c30 h30
or
Nn =
16πk 3 T 3 V
c30 h30
1
n4
(4.3)
If also, now, ν is written for the specific volume (V /N m) of the medium, then
the last relation, Eq. (4.3), becomes
v=
c30 h30
45c30 h30
=
16πmk 3 T 3 ζ(3) 8π 5 k 3 T 3 m
(4.4)
so that, from Eq. (3.10),
3kT
E=
=
m
8π 5 k 4
15c30 h30
5
vT 4
(4.5)
i .e .
E
= A0 T 4
v
where
A0 =
8π 5 k 4
15c30 h30
(4.6)
gives the radiation-density ’constant’.
It is now clear that, if there is a gas-like ethereal medium, such as has now
been derived, then c0 must be a special wave-speed, and h0 a special value of
some quantity, both of which may vary with position and time in the Universe.
The first and most obvious natural conclusion is that c0 is the wave-speed which
obtains in our galactic neighbourhood, at the present epoch, in the background
radiation field, i .e . the 2.7o K microwave background black-body radiation field;
and that h0 , or Planck’s ’constant’ is, similarly, the contemporary value in our
galactic neighbourhood of a quantity which may vary both with position and
time in the Universe.
On this basis, taking
h0 = 662.56 × 10−35 (kg)m/sec2
c0 = 0.3 × 109 m/sec, T0 = 2.7o K, ζ(3) = 1.202
and, of course, identifying the Universal constant k as Boltzmann’s constant,
k = 13.8054 × 10−24 (kg)m2 /sec2 · deg C
then it is found that the absolute mass quantum m, the mass of a unit
ether-particle, is
m = 0.497 × 10−39 (kg)
(4.7)
and
m = 0.552 × 10−39
(kg)
(4.8)
whilst v0 and p0 , the specific volume and pressure, respectively, of the local
contemporary background radiation field, are
v0 =
40h30
= 5.05 × 1030 m3 /(kg)
3π 5 c30 m4
(4.9)
1
3
= 0.198 × 10−30 (kg)/m
v0
(4.10)
or
ρ0 =
and
p0 = 13.37 × 10−15 (kg)/m · sec2
or
133.7 × 10−21 bars
(4.11)
It remains to summarise the implications of the results now obtained for a
particulate gas-like ethereal medium. Equation (4.2) differs vitally from Planck’s
quantum hypothesis and shows that it is unnecessary to make any hypothesis
of this kind. For the result, Eq. (4.2), is derived from observation and implies
6
that a particular frequency ν is not associated, as Planck postulated, with a
particular value of a continuously variable energy quantum h0 ν, but rather that
a particular frequency ν is associated with a particular value of energy per
unit mass , i .e . any particular frequency is associated with all those ether
particles, whatever their masses, which have a particular energy per unit mass,
= h0 ν/m, and which thus have different energies nh0 ν corresponding to their
different masses nm.
Equation (4.4) implies that νT 3 /c30 h30 or, by Eq. (4.6), νT 3 A0 is a universal
constant. Now, in an ideal gas with constant first adiabatic index 4/3, constant
νT 3 implies constant entropy so that Eq. (4.4) accords, as it must, with the
assumption made in Section 3 above that A0 (or c0 h0 ) is a function of entropy.
But further, if there is such an ethereal medium, the Universe must consist
of an expanding flow of ether in which matter is suspended. In this case, if there
are no ethereal shock waves, what are usually called ’world-lines’ in unsteady
fluid dynamics, but which are now become ’Universal-lines’, will be isentropic,
and Eq. (4.4) will then imply that, to an observer travelling with the ethereal
flow, A0 or c0 h0 will have a constant value for all time. And still further, if
not only are the Universal-lines isentropic, but the whole ethereal flow of the
Universe is homentropic, then Eq. (4.4) will imply that A0 or c0 h0 is a Universal
constant.
The calculated value, Eq. (4.7), for the absolute mass quantum m implies
that the mass of an electron is about 2 × 109 times that of a unit ether-particle;
whilst Eq. (3.10) implies that the constant specific heat at constant volume of
the ether is
3k
cv =
= 75 × 1015 m2 /sec2 · deg C
m
or about 18 × 1012 cal/g·deg C.
Equation (4.2) implies that, for instance, the frequency of red light, ν ≈ 12 ×
18
10 sec−1 is associated with an energy per unit mass
=
2
2
× 1024 m2 /sec
3
or
0.16 × 1021 cal/g
Equations (4.8) and (4.9) together imply that, in the local contemporary
background radiation field, there are, on average, at any given time, 0.359 × 109
ether-particles per cubic metre, or about 360 per cubic centimetre.
5 Historical note
It is of considerable historical interest to observe that de Broglie touched upon
the possibility of an infinite variety of light-quanta or photo-molecules, each of
energy nh0 ν, an integral multiple of Planck’s energy quantum h0 ν. He noted,4,5
that Planck’s distribution
could be expanded as an infinite series of terms in
ν 3 exp −nh0 ν/kT corresponding to Eq. (3.3) above, each term having the
form of Wien’s distribution. Einstein6 had derived the mean square of the fluctuation of energy per unit volume, from Planck’s distribution, as the sum of two
terms which were, respectively, the values which would have been obtained by
7
starting with Wien’s distribution or Rayleigh’s distribution, rather than with
Planck’s distribution. de Broglie also expanded Einstein’s result for the fluctuations as an infinite series, and found that it corresponded, term by term, with
the fluctuations calculated individually for the Wien-type terms in the expansion of Planck’s distribution. Thus the terms of each series could be regarded
as corresponding to energy quanta nh0 ν, and this suggested the possibility of
obtaining both Planck’s distribution and Einstein’s fluctuations on the basis of a
corpuscular or particulate theory of electromagnetic radiation, provided a suitably weighted mixture could be determined for these different corpuscles with
energies nh0 ν. Such a corpuscular theory had already been proposed earlier by
Wolfke.7
de Broglie does not appear to have pursued these suggestions further, or
attempted to determine whether a possible mixture of ’n-quanta’ existed. However, Bothe,8 apparently independently, since he does not refer to de Broglie,
went considerably further. He made use of Einstein’s hypothesis9 regarding the
emission and absorption of light by material molecules, in order to derive the
number of ’n-quanta’ in black-body radiation, and obtained the result, cf. Eq.
(3.1) above,
8π
−nh0 ν
1 ∂Nn (ν)
2
=
ν exp
(5.1)
V
∂ν
nc30
kT
On the hand, he proceeded to show that this led to the result, cf. Eq. (3.3)
above,
∞
∞
X
∂Nn (ν)
8πh0 V X 3
−nh0 ν
∂E(ν, T )
=
nh0 ν
=
ν exp
∂ν
∂ν
c30
kT
n=1
n=1
(5.2)
in agreement with Planck’s distribution as expanded in series form by de Broglie.
On the other hand he did not, surprisingly, integrate his result, Eq. (5.1), to
obtain quite simply
Z ∞
16πk 3 T 3 V
∂Nn (ν)
1
Nn =
dν =
(5.3)
3 h3
4
∂ν
c
n
0
0 0
in precise agreement with Eq. (4.3) above.
Neither de Broglie nor Bothe remarked upon the fact that their concept of
’n-quanta’ or photo-molecules implied a vital emendation of Planck’s quantum
hypothesis, in that it required an association of wave-frequency ν, not with a
quantity of energy, but with energy per n of such photo-molecules. And, since
their approach was based on such a quantum hypothesis and the associated
ideas of photons or energy-packets without mass, they both failed to recognise,
even though Bothe succeeded in working backwards to the solution given here,
that their concepts of a corpuscular theory, and their results, could be derived
by working forwards, without hypothesis, from the kinetic theory of a gas with
Maxwellian statistics.
8
References
1. Whittaker, Sir Edmund. A History of the Theories of Aether and Electricity, 2
volumes, Thomas Nelson, London and New York (1953).
2. Thornhill, C. K., Ministry of Defence RARDE Report, 2/73 (1973).
3. Swan, G. W., and Thornhill, C. K., Mech. Phys. solids, 22, 349 (1974).
4. de Broglie, Prince Louis Vistor, C.r., 175, 811 (1922).
5. de Broglie, Prince Louis Vistor, J. Phys. Radium. Paris, 3, 422 (1922).
6. Einstein, A., Phys. Z., 10, 185 (1909).
7. Wolfke, M., ibid, 22, 375 (1921).
8. Bothe, W., Phys., 20, 145 (1923).
9. Einstein, A., Phys. Z., 18, 121 (1917).
APPENDIX The specific energy distribution in a gas with an arbitrary number of degrees of molecular freedom
For a gas whose molecules have α degrees of freedom, let the component speeds
of the molecules be denoted by l1 , l2 , l3 , and let the contributions to the energy
per unit mass, of the remaining (α − 3) degrees of freedom be written as
1 2 1 2
1 2
2 l4 , 2 l5 , ..., 2 lα , so that
α
X
1 2
l
(A.1)
=
2 i
i=1
If N (l1 , l2 , l3 , ..., lα ) is the number of molecules whose specific energy components
lie in the ranges (0, l1 ), (0, l2 ), ..., (0, lα ), then, for Maxwellian statistics
!
α
X
∂ α N (l1 , l2 , l3 , ..., lα )
= K 0 exp −
δi li2
(A.2)
∂l1 ∂l2 ∂l3 ...∂lα
i=1
for some K 0 and δi . The mean square values of li are given by
R∞
exp −δi li2 li2 dli 1
2
0
= δi
Li = R ∞
2
exp −δi li2 dli
0
(A.3)
and, for equipartition of energy between all the α degrees of freedom, these must
all be equal, so that L2i = c2 /3, δi = 3/2c2 for all values of i, where c is the r.m.s.
speed of the molecules. Then,
!
α
X
∂ α N (l1 , l2 , l3 , ..., lα )
3li2
0
= K exp −
(A.4)
∂l1 ∂l2 ∂l3 ...∂lα
2c2
i=1
or
N (l1 , l2 , l3 , ..., lα ) = K
0
Z
l1
Z
l2
Z
l3
Z
...
0
0
0
0
lα
α
X
3li2
exp −
2c2
i=1
!
dl1 dl2 dl3 ...dlα
(A.5)
9
The multiple integral may be transformed by means of the generalised spherical
’polar’ co-ordinates,
l1 = u sin θ1 cos θ2 cos θ3 ... cos θα−1
l2 = u sin θ1 sin θ2 cos θ3 ... cos θα−1
l3 = u sin θ1 sin θ3 cos θ4 ... cos θα−1
..........
(A.6)
..........
lα−2 = u sin θ1 sin θα−2 cos θα−1
lα−1 = u sin θ1 sin θα−1
lα = u cos θ1
which satisfy
α
X
li2 = u2 = 2
(A.7)
i=1
This enables the integrations with respect to the θi to be performed, giving
N (u) or N (), the number of molecules lying in the ranges (0, u), (0, ). Thus
Z u
−3u2
N (u) = K
uα−1 exp
du
(A.8)
2c2
0
or
N () = 2
(α−2)
2
Z
K
(α−2)
2
0
exp
−3
c2
d
(A.9)
The constant K may be evaluated by integrating Eq. (A.9) from 0 to ∞, to
give the total number N of the molecules. Then, finally,
#
" 1
(α−2)
∂N ()
−3
3 2 αN
= α 1 2 exp
(A.10)
∂
c2
c Γ 2α
In the simplest case of a monatomic gas, for which α = 3 and the energy of the
atoms is entirely kinetic energy of motion, the general relation of Eq. (A.10)
reduces to
!
√
1
∂N ()
6 3N
−3
2 exp
=
(A.11)
1
∂
c2
π 2 c3
10
or, with u2 = 2
∂N (u)
=
∂u
!
√
3 6N
1
π 2 c3
u2 exp
−3u2
2c2
(A.12)
In this particular simple case, and in this case only, u is the speed of the atoms,
so that Eq. (A.12) must necessarily be the Maxwellian speed distribution. For a
gas whose molecules have six degrees of freedom, α = 6 and Eq. (A.10) reduces
to
27N
−3
∂N ()
2
=
exp
(A.13)
∂
2c6
c2
and this is the distribution, Eq. (3.1), required in Section 3 above, of the main
part of the paper.
11